On a surface, this appears to be a relatively simple data visualisation, displaying a map of historical and future (i.e. forecasted) near-surface temperature changes between 1850 and 2100. The data is presented in two different ways: on a global map, the temperature change is reflected by a color (the authors use a perceptually uniform color scale), but when you click on a particular location, you can also display a one-dimensional slice, temperature time series showing how this region of the world has warmed and could warm in the future.
Datasets used for this visualisation required some additional post-processing (for instance, historical data had to be blended with model forecasts), and converting from a relatively unfriendly geophysical format called NetCDF into small csv bits of data corresponding to 1x1 degree grid boxes. The frontend layer appears to be mostly d3.js.

What you see here is an animated globe with an overlay displaying weather forecast and other related data. The forecast data come from the Global Forecasting System (commonly called just "GFS"), but the visualisation uses also other data sources to display e.g. ocean wave height, particulate air pollution, or space aurorae.

The visualisation uses d3.js and a lot of homebrew code to draw weather data on HTML canvas. While some layers (e.g. wind vectors) appear to be animated, you can only see a static snapshot of the atmosphere at the particular time step, for instance at 00:00 UTC. You don't need to use the spherical projection, as the tool supports other fancy projections, including the Waterman butterfly (see below).

This type of climate data visualisation was made popular by dr Ed Hawkins from University of Reading. It show global temperature change (or other 1-dimensional dataset, typically with visible seasonal component) in radial coordinate system, with time dimension encoded by a color. The visualisation shown below is using JavaScript to reimagine the original animation.